Mathematicians and Music

On 6 September 1923 Raymond Clare Archibald of Brown University gave his Presidential Address to the Mathematical Association of America on Mathematicians and Music. His talk, which was delivered at Vassar College, Poughkeepsie, to a joint meeting of the Mathematical Association of America and the American Mathematical Society, was published in the American Mathematical Monthly in the January 1924 part. We give below a version of the first section of Archibald's talk:-

Mathematicians and Music

by R C Archibald, Brown University.

Mathematics and Music, the most sharply contrasted fields of intellectual activity which one can discover, and yet bound together, supporting one another as if they 'would demonstrate the hidden bond which draws together all activities of our mind, and which also in the revelations of artistic genius leads us to surmise unconscious expressions of a mysteriously active intelligence.

In such wise wrote [Helmholtz] one supremely competent to represent both musicians and mathematicians, the author of that monumental work, On the Sensations of Tone as a physiological basis for the Theory of Music.

"Bound together?" Yes! in regularity of vibrations, in relations of tones to one another in melodies and harmonies, in tone-colour, in rhythm, in the many varieties of musical form, in Fourier's series arising in discussion of vibrating strings and development of arbitrary functions, and in modern discussions of acoustics.

This suggests that the famous affirmation of Leibniz, "Music is a hidden exercise in arithmetic, of a mind unconscious of dealing with numbers," must be far from true if taken literally. But, in a very general conception of art and science, its verity may well be granted; for, in creating as in listening to music, there is no realization possible except by immediate and spontaneous appreciation of a multitude of relations of sound.

Other modes of expression and points of view were suggested by that great enthusiast to whom America owed much [Sylvester], him who called himself "the Mathematical Adam" because of the many mathematical terms he invented; for example, mathematic - to denote the science itself in the same way as we speak of logic, rhetoric or music, while the ordinary form is reserved for the applications of the science. He referred to the cultures of mathematics and music "not merely as having arithmetic for their common parent but as similar in their habits and affections." "May not Music be described," he wrote, "as the Mathematic of Sense, Mathematic as the Music of reason? the soul of each the same! Thus the musician feels Mathematic, the mathematician thinks Music, - Music the dream, Mathematic the working life, - each to receive its consummation from the other when the human intelligence, elevated to the perfect type, shall shine forth glorified in some future Mozart-Dirichlet, or Beethoven-Gauss - a union already not indistinctly foreshadowed in the genius and labours of a Helmholtz"!

But such intimacies in these cultures are not discoveries and imaginings of a later day. For two thousand years music was regarded as a mathematical science. Even in more recent times the mathematical dictionaries of Ozanam, Savérien, and Hutton, contain long articles on music and considerable space is devoted to the subject in Montucla's revised history, - which brings us to the threshold of the nineteenth century. It is, therefore, not surprising that many mathematicians wrote on musical matters. I shall presently consider these at some length. But certain other facts may first be reviewed.

The manner in which music, as an art, has played a part in the lives of some mathematicians is recorded in widely scattered sources. A few instances are as follows.

Maupertuis was a player on the flageolet and German guitar and won applause in the concert room for performance on the former. At different times William Herschel served as violinist, hautboyist, organist, conductor, and composer (one of his symphonies was published) before he gave himself up wholly to astronomy. Jacobi had a thorough appreciation of music. Grassmann was a piano player and composer, some of his three-part arrangements of Pomeranian folk-songs having been published; he was also a good singer and conducted a men's chorus for many years. János Bolyai's gifts as a violinist were exceptional and he is known to have been victorious in 13 consecutive duels where, in accordance with his stipulation, he had been allowed to play a violin solo after every two duels. As a flute player De Morgan excelled. The late G B Mathews knew music as thoroughly as most professional musicians; his copies of Gauss and Bach were placed together on the same shelf. It was with good music that Poincaré best liked to occupy his periods of leisure. The famous concerts of chamber music held at the home of Emile Lemoine during half a century exerted a great influence on the musical life of Paris. And in America we have only to recall colleagues in the mathematics departments of the Universities of California, Chicago and Iowa, and of Cornell University, who are, to use Shakespeare's phrase, "cunning in music and mathematics."

While Friedrich T Schubert, the Russian astronomer and mathematician, played the piano, flute, and violin in an equally masterly fashion, his great-grand-daughter Sophie Kovalevskaya was devoid of musical talent; but she is said to have expressed her willingness to part with her talent for mathematics could she thereby become able to sing. Abel had no interest in music as such, but only for the mathematical problems it suggested. His close attention to a performer at a piano was once explained by the fact that he sought to find a relation between the number of times that each key was struck by each finger of the player. Lagrange welcomed music at a reception because he could by the fourth measure become oblivious to his surroundings and thus work out mathematical problems; for him the most beautiful musical work was that to which he owed the happiest mathematical inspirations. Dirichlet seemed to be sensible to the charms of music in a similar manner.

Such are a few instances, which could be considerably multiplied, of the relation of mathematicians to the art of music:-

that gentlier on the spirit lies
Than tir'd eyelids upon tir'd eyes.

They suggest the accuracy of at least a part of the following observations of Möbius in his book on mathematical abilities:-

Musical mathematicians are frequent ... but there are wholly unmusical mathematicians and many more musicians without any mathematical capability.

That there are musicians with some mathematical ability will be granted when we recall, not only that Henderson, the prominent New York music critic and the author of many works on musical topics, has written a little book on navigation, but also that the late Sergei Tanaieff, pupil of Rubenstein and Tchaikovsky, successor of the latter as professor of composition and instrumentation at the Moscow Conservatory of Music, and one of the most prominent of modern Russian composers, found algebraic symbolism and formulae of fundamental importance in his lectures and work on counterpoint.

A question which has interested more than one group of inquirers is: Can one establish any relationship between mathematical and musical abilities? Within the past year two Jena professors, Haecker and Ziehen, published the results of an elaborate inquiry as to the inheritance of musical abilities in musical families. As a by-product of the inquiry they arrived at the result that in only about 2 per cent of the cases considered was there any appreciable correlation between talent for music and talent for mathematics; they found also that the percentage of males lacking in talent for music but showing a talent for mathematics was comparatively high, about 13 per cent. At the Eugenics Record Office of Cold Spring Harbor, Long Island, there has been collected a considerable body of data upon which a study of the correlation of mathematical and musical abilities could be based. It will be interesting to see if the conclusions of Haecker and Ziehen are here checked, and also if some results are found as to the extent to which musical abilities are present in a group of mathematicians.